Mogens Steffensen

Bankruptcy, Counterparty Risk, and Contagion

This paper provides a unifying framework for the modeling of various types of credit risks such as contagion effects. We argue that Markov chains can efficiently be used to tackle these problems. However, our approach is not limited to pricing problems with contagion. Other applications include the modeling of a more sophisticated default process of a firm. On the theoretical side, we derive pricing formulas for three building blocks that are generalizations of contingent claims studied in Lando (1998). These claims can be thought of as atoms forming the basis for all credit risky payments. Furthermore, we demonstrate that, in general, all contingent claims exposed to credit risk satisfy a system of partial differential equations. This is the key result to calculate prices of credit risky claims explicitly and efficiently.

Intervention options in life insurance

Abstract We deal with the intervention options of the policy holder in life insurance. To these options belong the surrender and the free policy (paid-up policy) options. Our approach is to let payments be driven by processes in which the policy holder is allowed to intervene. The main result is a quasi-variational inequality describing the market reserve on an insurance contract taking into account intervention options. The quasi-variational inequality generalizes Thiele’s differential equation used for calculation of reserves on a policy without intervention options. It also generalizes the classical variational inequality used for calculation of the price of an American option.

A no arbitrage approach to Thiele's differential equation

Abstract The multi-state life insurance contract is reconsidered in a framework of securitization where insurance claims may be priced by the principle of no arbitrage. This way a generalized version of Thiele’s differential equation (TDE) is obtained for insurance contracts linked to indices, possibly marketed securities. The equation is exemplified by a traditional policy, a simple unit-linked policy and a path-dependent unit-linked policy.

Household consumption, investment and life insurance

This paper develops a continuous-time Markov model for utility optimization of households. The household optimizes expected future utility from consumption by controlling consumption, investments and purchase of life insurance for each person in the household. The optimal controls are investigated in the special case of a two-person household, and we present graphics illustrating how differences between the two persons affect the controls.

OPTIMAL CONSUMPTION AND INSURANCE: A CONTINUOUS-TIME MARKOV CHAIN APPROACH

Personal financial decision making plays an important role in modern finance. Decision problems about consumption and insurance are in this article modelled in a continuous-time multi-state Markovian framework. The optimal solution is derived and studied. The model, the problem, and its solution are exemplified by two special cases: In one model the individual takes optimal positions against the risk of dying; in another model the individual takes optimal positions against the risk of losing income as a consequence of disability or unemployment.

A Dynamic Programming Approach to Constrained Portfolios

This paper studies constrained portfolio problems that may involve constraints on the probability or the expected size of a shortfall of wealth or consumption. Our first contribution is that we solve the problems by dynamic programming, which is in contrast to the existing literature that applies the martingale method. More precisely, we construct the non-separable value function by formalizing the optimal constrained terminal wealth to be a (conjectured) contingent claim on the optimal non-constrained terminal wealth. This is relevant by itself, but also opens up the opportunity to derive new solutions to constrained problems. As a second contribution, we thus derive new results for non-strict constraints on the shortfall of intermediate wealth and/or consumption.

Consumption-Portfolio Optimization with Recursive Utility in Incomplete Markets

In an incomplete market, we study the optimal consumption-portfolio decision of an investor with recursive preferences of Epstein–Zin type. Applying a classical dynamic programming approach, we formulate the associated Hamilton–Jacobi–Bellman equation and provide a suitable verification theorem. The proof of this verification theorem is complicated by the fact that the Epstein–Zin aggregator is non-Lipschitz, so standard verification results (e.g. in Duffie and Epstein, Econometrica 60, 393–394, 1992) are not applicable. We provide new explicit solutions to the Bellman equation with Epstein–Zin preferences in an incomplete market for non-unit elasticity of intertemporal substitution (EIS) and apply our verification result to prove that they solve the consumption-investment problem. We also compare our exact solutions to the Campbell–Shiller approximation and assess its accuracy.

On Merton’s Problem for Life Insurers

This paper deals with optimal investment and redistribution of the free reserves connected to life and pension insurance contracts in form of dividends and bonus. Formulated appropriately this problem can be viewed as a modification of Merton’s problem of optimal consumption and investment with a very particular form of consumption and utility hereof. Both are linked to a finite state Markov chain. We distinguish between utility optimization of dividends, where a semi-explicit result is obtained, and utility optimization of bonus payments. The latter connects to the financial notion of durable goods and allows for an explicit solution only in very special cases.

Worst-case-optimal dynamic reinsurance for large claims

We control the surplus process of a non-life insurance company by dynamic proportional reinsurance. The objective is to maximize expected (utility of the) surplus under the worst-case claim development. In the large claim case with a worst-case upper limit on claim numbers and claim sizes, we find the optimal reinsurance strategy in a differential game setting where the insurance company plays against mother nature. We analyze the resulting strategy and illustrate its characteristics numerically. A crucial feature of our result is that the optimal strategy is robust to claim number and size modeling and robust to the choice of utility function. This robustness makes a strong case for our approach. Numerical examples illustrate the characteristics of the new approach. We analyze the optimal strategy, e.g. in terms of the more conventional, in the insurance context, objective of minimizing the probability of ruin. Finally, we calculate the intrinsic risk-free return of the model and we show that the principle of Markowitz—don’t put all your eggs in one basket—does not hold in this setting.

Deterministic mean-variance-optimal consumption and investment

In dynamic optimal consumption–investment problems one typically aims to find an optimal control from the set of adapted processes. This is also the natural starting point in case of a mean-variance objective. In contrast, we solve the optimization problem with the special feature that the consumption rate and the investment proportion are constrained to be deterministic processes. As a result we get rid of a series of unwanted features of the stochastic solution including diffusive consumption, satisfaction points and consistency problems. Deterministic strategies typically appear in unit-linked life insurance contracts, where the life-cycle investment strategy is age dependent but wealth independent. We explain how optimal deterministic strategies can be found numerically and present an example from life insurance where we compare the optimal solution with suboptimal deterministic strategies derived from the stochastic solution.